DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Type Iia Potential Gauge Transformations

Physical scope

Local potential gauge transformations in the source-free massless Hamilton convention. The verification is kinematic and does not establish a global gauge group, gerbe data, quantization, or source transformation law.

Definition

On a smooth ten-dimensional local patch $U$ , take smooth real potentials $B_{2} \in Ω^{2} (U)$ , $C_{1} \in Ω^{1} (U)$ , $C_{3} \in Ω^{3} (U)$ with $H_{3} = d B_{2}$ and $d H_{3} = 0$ , and let $ζ_{1}, Λ_{0}, Λ_{2}$ be smooth one-, zero-, and two-form gauge parameters. The Hamilton massless type IIA potential transformations are $B_{2} ⟼ B_{2} + d ζ_{1}, C_{1} ⟼ C_{1} + d Λ_{0}, C_{3} ⟼ C_{3} + d Λ_{2} + H_{3} Λ_{0} .$ The gauge-invariant local field strengths are $H_{3} = d B_{2}$ , $G_{2} = d C_{1}$ , and ${\tilde{G}}_{4} = d C_{3} - C_{1} \land H_{3}$ . The auxiliary strength $G_{4} : = d C_{3}$ transforms by $G_{4} ⟼ G_{4} - H_{3} \land d Λ_{0}$ and is generally not invariant. In the length-valued coordinates of the adopted action, potential components are dimensionless; gauge-parameter components have dimension length so that each displayed potential shift is dimensionless.

Definition references

M. J. D. Hamilton, §6.1, Definition 6.1

Type Iia Two Derivative Supergravity Action

Facts & Assumptions

Given: Smooth gauge parameters, $d H_{3} = 0$ , and the potential/field-strength conventions stated above.

[A1]

Supergravity Conventions And Normalizations (current batch-56 carrier and current ready Step 1 record inspected; its stated hypotheses match this use): ordinary exterior differentiation and wedge products are used in Lorentzian ten dimensions.

[A2]

Type Iia Two Derivative Supergravity Action adopts the local definitions of $H_{3}, G_{2}, G_{4}, {\tilde{G}}_{4}$ used here.

[F1]

The exterior derivative is a graded derivation: $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β$ .

[F2]

The exterior derivative squares to zero: $d^{2} = 0$ on smooth forms.

Verification

1.1givenA2F2algebra

Under $B_{2} \mapsto B_{2} + d ζ_{1}$ , $H_{3} \mapsto H_{3} + d^{2} ζ_{1} = H_{3}$ by [F2]. Under $C_{1} \mapsto C_{1} + d Λ_{0}$ , $G_{2} \mapsto G_{2} + d^{2} Λ_{0} = G_{2}$ .

2.1givenA1A2F1step 1.1algebra

With $C_{3} \mapsto C_{3} + d Λ_{2} + H_{3} Λ_{0}$ , the change in $G_{4}$ is $d (H_{3} Λ_{0}) = d H_{3} Λ_{0} - H_{3} \land d Λ_{0} = - H_{3} \land d Λ_{0}$ , by [F1] and $d H_{3} = 0$ . The change in $C_{1} \land H_{3}$ is $d Λ_{0} \land H_{3} = - H_{3} \land d Λ_{0}$ , since degrees 1 and 3 anticommute in [A1]. Thus ${\tilde{G}}_{4}$ is unchanged.

3.1givenA2step 1.1step 2.1∎

The transformation leaves $H_{3}, G_{2}, {\tilde{G}}_{4}$ invariant by steps 1.1 and 2.1; step 2.1 also gives the stated transformation of $G_{4}$ . This is the stated local gauge invariance of the field-strength data; it does not by itself assert invariance of a global action with boundary or source terms.

Depends on

Dependency tree · 2 levels

Sources

M. J. D. Hamilton, The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation